Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C4×#3556 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4);(5,6,10,8)(7,13,15,11)(9,16,17,12)(14,18,20,19)(21,22,26,30)(23,28,24,27)(25,33,34,29)(31,35,32,36);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,22)(23,28)(24,27)(25,29)(26,30)(31,36)(32,35)(33,34);(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(5,8,10,6)(7,11,15,13)(9,16,17,12)(14,18,20,19)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,25,26,34)(22,29,30,33)(23,31,24,32)(27,35,28,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533397);